Measurement of isobarbic specific heat of He4 and HeHe2 mixtures

The specific heat, Cp, of He4 and the binary system He4-normal hydrogen has been studied using the flow calorimeter method. A short description of the apparatus and estimation o f errors is presented. The specific heat of He4 under 0.507, 1.502, 2.533 and 4.053 MPa isobars in the temperature range 15-40 K and for I MPa isobar in the temperature range 15-100 K was obtained. The specific heat of He-H2 mixtures was measured at temperatures of 15-40 K and at pressures 0.507, 1.013, 2.503, 4.053 MPa, in the range of He weight ratios from O. 106 to O. 988.

Measurement of isobarbic specific heat of He 4 and H e - He 2 mixtures V . K . Y a r o u n t s e v and V . A . Medvedev

The wide use of materials with low boiling points in cryogenic technology accounts for the existing interest in their thermophysical properties. Many papers have been devoted to the properties of Helium and H2. A number of P-V-T measurement data has been generalized in equations of state and the detailed tables of their properties computed. It was repeatedly mentioned that one of the best criteria of the equations' reliability is its agreement with the value of the specific heat, calculated, for example, under isobaric conditions, with direct experimental results. Isobaric, specific heat, experimental data for H2 are available at parameters close to the saturation lines. 3 Cp measurements for He in the 'hydrogen' temperature range have not, as yet, been carried out. As far as He-H2 mixtures are concerned a number of papers have been recently published. In particular, the liquidvapour phase equilibrium of helium- normal H2 and heliumpara-H2 systems have been studied. 4'~ The compressibility measurements, 6 and Joule-Thompson effect measurements 7 have been carried out and the second virial coefficient measured at 20.4 K. 8 However, there are still no reliable compressibility and caloric data of the system in the range of parameters close to the saturation line. Apparatus and procedure

T h e a u t h o r s m a y be c o n t a c t e d via J.L. Olsen, Eidgenossische Techische Hochschute, L a b o r a t o r i u m f u r FestkOrper Physik, HOnggerberg C H - 8 0 4 9 Z 0 r i c h , S w i t z e r l a n d . Received 13 January 1977.

. SEPTEMBER

RI

TBX

K

R2

TBblX

An adiabatic flow calorimeter (analogous to the one documented, 9'~° but modified for operation with substances both in one-phase and two-phase states) had been used for measuring the specific heat Cp (Fig. 1). A membrane compressor, 13, (Fig. 2) circulates the substance under test in the unit. Constant gas pressure and flow rate in the system are maintained by means of a two-stage bellows pressure regulator at the calorimeter inlet 7,8 and by means of low pressure regulator, 14, at the calorimeter outlet. Preset pressure in the calorimeter is maintained by the electromechanical automatic monitoring system. The pressure is initially set by the percision piston gauge 5 MP-60 (degree

CRYOGENICS

T

1977

R4

Fig. 1 Cross-section o f the apparatus. 1,2 - gas inlet and outlet; 3,4 - tubes f o r pressure measurement; 5,6 - radiation shield; 7 - thermal decoupling; 8 - boiler; T -- t h e r m o s t a t , T B X - inlet and T B b / X -- o u t l e t t h e r m o m e t e r ; R I , R 2 , R3, R 4 - electric heaters

509

1 - differential pressure gauge; 2 - cryostat; 3 - throttle; 4 - bellows separator with inductive Fig. 2 Flow diagram of the calorimeter. sensor; 5 - piston gauge; 6 - manostate; 7.8 - stages of bellows pressure regulator; 9 - motor; 10 - adsorber; 11 - vessels with pure gases; 12 - bypass; 13 - membrane compressor; 14 - manostat; 15 - gasmeter; 16 - electric valves; 17 - gasmeter vessel; 18 - absolute pressure gauge; 19 - thermostat; 20 - capillars for pressure levelling; 21 - analyser tubes; 22 - pure gas vessel

of accuracy 0.02) and maintained by the operating unit which is the bellows reference pressure variator of the second stage regulator’ with motor 9. Gas flow rate, rri, is measured, with an error not exceeding 0.07%, by a volumetric type precision gasometer, 15. The weight fraction x of He in the mixture is determined by an interferometric analyser, 21. The difference between the pure component refraction coefficient and the mixture refraction coefficient determines the composition. The stability with which nominal temperature conditions are maintained iswithin 5 x 10m4K. The heat generated, N, in the calorimeter by the electric heater is determined by the potentiometric method. The temperature increase of the substance, AT, in the calorimeter is also determined by the potentiometric method from the resistance of two precision platinum thermometers, jointly calibrated.” The calorimeter design made it possible to reduce heat leak to the negligible value (- 0.01%). The pressure drop in calorimeter, measured by the mercury differential gauge, was - 100 Pa, consequently the throttling temperature effect therefore did not exceed 1O3K. So the simple relation

c, = -

N

niAT

510

Table 1.

x 10-3Wt

20-80 0.5-2

0.02%

K

0.1%

2.10-3-lo-*g s-’

0.07%

15-100

0.01

05-4

K

K

0.02%

MPa

o-1

0.003

may be used to calculate the average specific heat to the accuracy required. The range of the basic values measured and their respective errors are given in Table 1. The total mean systematic error of the specific heat measurement is from 0.25% on the sloping areas of isobars to 1% near the maximum. Specific heat of helium The data of He C, were obtained on the isobars 0.5 15, 1.522, 2.5 10 and 4.058 MPa at temperatures from 15 to 40 K.” These results with due correction introduced later are given in Table 2. To determine the metrological parameters of the apparatus, a detailed investigation of the specific heat of He on 1 MPa

CRYOGENICS.

SEPTEMBER

1977

isobar was carried out in a wide temperature range (up to 100 K).” Fifty four values of C, were obtained and described using the method of the least squares

p = I 013 MPa

6 c,

=

c

J R-’ K-l-

ai.T’,

i=-_l

The standard deviation of experimental data from the ‘smooth’ theoretical curve equals 0.01 J g-’ K-’ . For the chosen parameters there is but a slight difference between the specific heat of helium and an ideal gas, so that it can be calculated with good accuracy by the thermal equation of state. In fact, the calculated results’ comply with our data at 1 MPa within 0.2%. Results for He-H2

70

(2)

mixtures

The mixtures were prepared from high purity He (99.99% He) and the commercial Hz which was processed by adsorption purification. The specific heat of these mixtures was measured on isobars at 0.507, 1 .013,2.533, 4.13 MPa pressures the composition of the mixture being constant (x = const, x - weight of helium in the mixture). The main measurements were made at other concentrations to define, with better accuracy, the relationship between specific heat and composition for parameters close to

60

x--x

=0936

+-x

=0900

50

; Y ; o 3

40

30

20

IO

-_

20

25

Fig. 4

C,(T)

- diagram at p

=

1.013

MPa

p =0.507MPa

70

60

o

- x=O.602(p=O31 MPa

0

-x =0.374

a

-x=0400

.

-x=0632

x

-x=0936

Table 2. P,MPa

T,K

4.058

15.59 16.89 18.49 19.75 20.92 23.10 25.63 26.87 28.40 30.35 33.39 35.68 38.39 40.88 15.87 17.26 19.14 21.13 23.94 25.33 28.01 29.92

2.514

2.509

Fig. 3

35

30

T, K

)5 45

C,(T)

- diagram at p

CRYOGENICS.SEPTEMBER

=

0.507

MPa

1977

CP

Jg-IK-, 6.01 6.04 6.04 6.035 6.00 5.95 5.87 5.84 5.80 5.76 5.69 5.63 5.59 5.54 6.34 6.24 6.125 5.99 5.85 5.77 5.68 5.63

P,MPa

T.K

2.509

33.06 35.63 38.00 40.56 15.83 17.82 19.22 21.02 22.38 25.44 28.61 31.48 34.88 37.61 40.57 15.40 17.09 18.58 20.46 25.82 33.39 39.35

1.522

0.515

CP Jg-'K-l 5.56 5.52 5.50 5.456 6.32 6.12 6.02 5.77 5.71 5.61 5.53 5.50 5.43 5.40 5.38 5.54 5.48 5.46 5.39 5.34 5.30 5.26

511

-1

p = 2.533 MPo 4 -x=0106 . -x = 0 270 _

p = 0.507

0 -x=0344

MPo

n

70

0 -x=0374 n . o .

-x=0484 -x=0670 -x=0682 -x=0893

60

x -x=0936

+ -x=0988

,;

Y

T 0 -2

1’

I-

I-

IO

T. K Fig. 5

C,(T)

- diagram at p

=

2.533

l t

MPa

I

I

I

I

02

04

06

08

J

IO

x

p=4053MPo Fig. 7

o - x = 0.106

CP(x)

- diagram at p =

0.507

MPa

D - x = 0 344 0 - x = 0477

S&ration line. The results are given in the Appendix (Table 4). C,,(T) at x = const for four pressures is shown in Figs 3-6. The dashed line denotes the leaps of the specific i:eat in the phase transfer point: two-phase region-gas or two-phase region-liquid (curve x = 0.106 at P = 2.533 and 4.053 MPa). The isotherms C,(x) of He-H2 system at the same pressures were plotted in Figs 7-10. The data of phase equilibrium4 were used to determine the phase transfer points. The values C, for pure hydrogen were taken from (2) because as it has been shown’3 and as calibration tests showed, there is no actual difference between C, of parahydrogen and normal hydrogen at our parameters.

n-x:0400 o- x =0602 A - x = 0.093 x-x=0936 + - x = 0.980

Discussion of results As can be seen from Figs 7- 10, specific heat isotherms in the two-phase region become straight lines. At the transfer points: liquid-two-phase region and two-phase region-gas the specific heat changes by leaps. The only exception being one point on the left part of the boundary curve (for low pressures), where the isotherm is broken. We shall consider the two-phase system enthalpy it to be additive to the mass of the phases: I 20

I __l-_ 25

30 T,

Fig. 6

512

C,(T)

- diagram at p

=

K

4.053

MPa

35

40

H = H,m,

assuming

+ Hzmz

where H .- specific enthalpy enthalpy of phase 1 (liquid),

(3) of the system, H, -- specific Hz - specific enthalpy of

CRYOGENICS.

SEPTEMBER

1977

80

p = 2 533

70

p = I013

MPa

MPo

60 1 I

25

/

50

7 Y L -J

40

30

\

20

A

IO

A

c IA 02

Fig. 9 Fig. 8

CPh)

-diagram

at p

=

1.013

C,(x)

- diagram

at p

I

I

I

04

06

00

x

=

2.533

MPa

MPa

phase 2 (gas), ml -- liquid mass, m2 - gas mass,.

ml + mz = 1

3ol

(4)

We chose pressure P, temperature T and the fraction of weight of one of the mixture components x to be the independent variables which determine the state of the system. Differentiating (3) we obtain:

25

c

p

053

MPo

20

7 Y ;

0, -3

15

IO

5

where i = 1, 2 is the phase index. Taking into account the fact that ti/dT = 0 and dP/dT = 0, (P, T, x - independent variables) and

I

I

I

06

dmr ------= dT

-

CRYOGENICS.

x

dm2 -dT

SEPTEMBER

(6)

1977

Fig.

08

and this determines the sign of the specific heat leap (CP - Cpl )’ on the left boundary curve at those pressures. Using (9) and (10) and the values (CP - C,, )” obtained from the C,(x) diagrams, the values of the specific heat leaps (Cr, - C,,)’ on the boundary curve of the liquid-twophase region were calculated. The necessary derivative values dx’/dT and dx”/dT are determined by differentiating the isobars of phaes equilibrium. A comparison of the obtained data with the theoretical calculations was carried out.14 The method based on the use of the properties of the pure components:

l-p 2-p 3- p 4-p

C,(P,T) =

=0.5O?MPa = 1.013 MPo = 2533 MPa = 4.053MPb

c

Cpi
i

+ 2x*x2 *PT.

II

d2Blz

(11)

dT2

where Cp(P, T) - heat capacity of mixture at P, T,

I

I

I

I

0.2

0.4

0.6

08

Cpi(Pi, T) - ith component Pi = PXiy

I Fig. 11

Phase diagram of system He-Hz

we obtain, for constant

c,

Xi - ith component

’

of specific heat at pressure

weight fraction,

pressure:

p = 4053

C,imi

=

(HI --Hz)

+ 2

(7)

MPa

-

Our data

-

Theoretical data

i

where C, is the two-phase system specific heat..If we express mi through x using the lever rule then:

c,

=

c,,

xs

+ -

+

dx’(x”-x) dT (x” -

For P, coordinates.

cp2

xs

x’

25

(x-x’) (HI

x’)

(8)

the

We the value of leaps at transfer points. From the liquid side at

ccp-

-Hz)

(8)

in phase x’

$gH;,,-;

and from the gas side at x = x’

(C, -Cp2)”

= g

s

(10) I

06

06

From the phase diagram of the He-H2 systen, (Fig. 11) one can see that the derivative dx’/dT at pressures 0.507 and 1.013 MPa passes through zero with the temperature increase

514

8 Comparison of our data with theory.” Fig. 12 __- theoretically calculated data

CRYOGENICS.

-o-

our data;

SEPTEMBER

1977

Table 3. i

-1

ai

0

38.1516

2.86942

3

i

ai

-5.95313

x 1o-6

5

6.91755 x lo-*

0.8884 x 10-I'

6 7

- was calculated from Lennard-Johnes potential

(15) 8

The comparison of results for three temperatures at 4.053 MPa are shown in Fig. 12. The larges deviations are observed on the calculated curves near the maxima. At He concentration x > 0.8 the difference does not exceed 2%.

References

9 10

11

McCarty, R.D. Thermophysical properties of helium4 from 2 to 1500 K with pressures to 1000 atmospheres, NBS Technical Notes 631 (1972) Roder, H.M., Weber, L.k, Goodwin, R.D. Thermodynamical and related properties of parahydrogen from triple point to 100 K at pressures to 340 atmospheres, NBS Monograf (1965) 94 Medvedev, V.A. Measurements of hydrogen isobaric heat capacity at temperatures 20-70 K and at pressures 2-500 atm, Diss Moscow (1968) Street, W.B., Sonntag, R.E., Wylen van G.J. Liquid - vapour equilibrium in system normal hydrogen-helium, J Chem Phys 40 (1964) 1390 Sonntag, R.E., Wylen van, G.J., Grain, R.W. Liquid-vapour equilibrium in the system parahydrogen-helium, J C/rem Phys 41 (1964)

2

5.47723 x IO-'

4

B12 - mixed second virial coefficient. -d’B,z dT2

1

12

13

14

15

-3.12491

x 1o-4

6 -2.2208

x IO-"

Dobrovolsky, O.A. Density measurements of nitrogen, hydrogen, meth ane, helium and helium-hydrogen mictures, Thesis (Moscow, 1967) Ostronov, M.G., Bolahakov, P.E., Galperin, 1.1. Joule-Thompson effect in gas mixtures at low temperatures, Journal of Physical Chemistry (Russ) 42 (1968) 5 Knaap, H.F.P., Knoester, M., Varekamp, F.H., Beenakker, J.J.M. The second virial coefficient of binary mixtures of the hydrogen isotopes and helium at 20.4 K, Physica dee 126 8 (1960) Yarountsev, V.K., Medvedev, V.A. Experimental apparatus for measurement isobaric heat capacity of cryogenic gases and liquids,Merrologiu and Measurement Technique 3 (1974) 33-36 Medvedev, VA., Yarountsev, V.K., Dedikov, LA., Precision flow calorimeter for hydrogen temperature range, Rep 6th All-Union conference on the calorimetry (Tbilisi 1973) Medvedev, V.A., Yarountsev, V.K. Helium-4 isobaric heat capacity at temperatures 16-40 K and pressures up to 40 atm. Physical constants and properties of the materials. Thermophysical properties of substances and materials 7 (1972) Medvedev, V.A., Yarountsev, V.K. Helium4 isobaric heat capacity at pressure 1 MPa in the temperature range 15- 100 K. Thermometric and thermophysical measurements research at low temperatures, VNIIFTRI Rept 21,Sl (Moscow 1975) Eselson, B.N., Blagoy, UP., et al. Liquid and solid hydrogen properties. The State Service of Standardization and Reference, Reference Review 1 (Moscow 1969) Altunin, V.V., Bondarenko, V.F., Kuznetaov, D.O. Pehnomenologic method for calculations of compressed gas mixtures enthalpy, Thermophysics of high temperatures, 12 (1974) 6 Hirschfelda, J.O., Curtis, C.F., Bird, R.B. Molecular theory of gases and liquids (New York 1954)

Appendix iable 4.

T, K

C

J&&l

T, K

c

J&l

x = 0.106

27.90

17.32

p = 2.533 MPa

30.62

23.11

33.65

23.46

22.60 25.10 29.25 30.78 32.70

10.84 12.68 15.63 18.00 22.66

x = 0.293

T, K

17.52

C ,&-I

10.76

p = 2.537 MPa

22.73

32.20 66.42

23.06

11.82

24.50 25.09

105.45

25.75

14.20

27.93

12.58

16.70

31.56

10.58

22.54 24.76 24.84

33.55 36.63 39.61 p = 7.03OMPa

10.28 9.96 9.65

24.60 28.29

13.49 19.70

20.17 22.51

11.95 16.14

1977

14.11

39.40 x = 0.4546

21.43 19.29

CRYOGENlCS.SEPTEMBER

17.86 15.25

15.44

27.46

27.02 27.33 29.144

35.20

28.53

p = 3.028 MPa

c

,,/,A

37.67

19.80

29.73 31.68 x = 0.344 p = 2.533 MPa

11.00 11.81 11.63 12.76

K

22.36

p = 4.053 MPa

10.75 10.85

T,

16.49

24.01 x = 0.4768

8.45 13.85

P = 4.053 MPa

21.57 25.81 27.79 27.80 29.20

10.58 13.72 16.04 16.05 18.52

515

(Table 4 continued) C P’

T.

J g-’ K-’

K

T, K 29.145

12.77

30.64

31.47

14.25

p

=

C

T J

K 26.82 4.053

MPa

J

K

24.84

24.34

27.14

45.76

33.59

15.78

28.38

79.02

35.15

17.12

23.48

11.62

30.92

13.56

36.33

18.02

27.30

14.53

34.18

12.10

38.26

19.26

31.30

14.30

36.50

11.47

x = 0.270

34.48

15.42

39.50

p = 2.533 MPa

39.28

15.05

p

=

c

T.

,-:,-I

,-:,-I

11.01 2.529

MPa

x = 0.374 22.14

11.23

24.80

13.14

x = 0.480 p

=

0.5284

MPa

p

=

0.5127MPa

25.66

34.31

27.12

62.78

p

=

21.32

32.41

2.516MPa

p

=

4.088

MPa

26.36

27.26

20.90

28.16

23.26

15.27

32.88

9.46

17.10

10.93

28.84

19.61

x = 0,670

18.88

14.47

17.42

8.76

30.29

14.50

P = 2.533

21.26

24.20

19.24

9.93

32.45

14.11

23.55 24.57

50.81

21.13 23.12

11.09

34.46

13.91

20.75

10.98

82.81

12.98

36.47

13.49

21.62

11.78

26.51

10.51

24.64

14.82

38.49

12.80

23.44

13.65

28.54

10.36

32.90

14.83

x = 0.632

31.10

10.16

34.48

13.63

p

31.57

10.35

36.50

12.52

33.64

10.22

11.54

34.66 37.57

9.50 9.36

39.09 p = 4.0793

39.59

9.19

x = 0.484 p

=

7.0193MPa

MPa

=

0.5066

MPa

MPa

26.08

17.90

29.64

10.44 8.94

35.10 x = 0.682

17.00 21.04

11.83 26.41

22.20

41.52

25.01

8.94 9.01

17.63 19.29

18.44 30.13

p

0.310MPa

=

18.61

8.43

20.41 21.52

9.65 10.69

30.50

8.68

20.70

51.19

23.46

12.17

30.53

8,78

21.30

64.96

8.42

23.43 p = 1.021 MPa

26.25

16.30

8.61

25.53

14.06

36.58

18.14

10.21

27.46

16.65

x = 0.658 p = 2.533

8.47

19.95

12.57

29.33

20.66

21.79

15.98

30.34

14.33

17.40

10.27

21.99

16.54

31.37

14.10

25.37

16.60

18.16

11.19

23.36 21.88 23.76

20.58 19.00

33.34 22.49

13.90 11.20

27.25 21.21

20.90 18.72

19.97 32.57

14.11 6.81

27.18

24.43

12.96

22.33

6.65

36.48

6.63

39.69 9.72

25.82

14.61

23.21

6.64

6.49

8.93

26.79 27.02

16.02 16.26

24.09 25.58

6.69 6.83

39.36 x = 0.936

8.48

27.81

17.77

30.60

6.57

29.65

10.57

p

32.42

10.66

25.23 26.92 29.68 33.86 p

=

2.506

MPa

=

2.520

MPa

p

=

0.5137

MPa

23.08

5.91

25.27

5.90

16.92

8.80

36.37

9.97

16.04

8.24

28.74

5.86

18.54

9.72

39.46

9.52

18.66

9.83

32.35

5.816

19.65

10.63

x = 0.893

19.95

10.85

37.97

21.10

11.87

p

20.24

11.13

p

25.02

7.13 6.93 6.72

=

1.001 MPa

22.60

13.40

24.11

15.63

16.04

8.67

25.48

18.30

16.57

9.24

26.67 30.18

26.37 27.33

20.56 23.62

17.04 17.46

9.73

P

10.26

27.80

25.13

18.00

11.18

516

=

16.90

4.058

MPa 7.85

=

5.816 0.9950

MPa

22.90

6.166

25.57

6.086

28.11

6.01

30.10

5.96

32.94

5.92

CRYOGENICS.

SEPTEMBER

1977

(Table

4 continued)

29.94

11.08

18.87

12.41

19.54

35.19

9.57

18.89

12.57

21.93

19.82

14.43

25.13

7.24

39.22

19.84

25.60

7.23

p

27.38

7.13

29.53

7.03

23.35

6.62 6.46

5.75

26.29 p = 4.060 MPa 21.66

p

=

4.058

MPa

18.30

8.47

20.77

14.91 17.36

20.01

9.45

20.78

17.39

20.98 28.61

9.96 6.34

29.21

31.37

6.24

x = 0.988

34.16 37.75

6.15 6.07

p

6.00

17.54 19.72 22.40

40.40 p

=

4.043MPa

=

6.53

34.51 p

=

9.00 10.69

33.09

5.92

35.39

5.89

=

5.83 2.541 MPa

7.0163

2.450 MPa 17.22

5.99

24.46

6.30 6.16

6.45

19.93

5.83

26.74

6.09

6.30 6.12

25.27 28.27

5.67 5.61

28.28 30.22

6.02 5.96

23.31

6.73

25.30

5.99

30.32

5.61

35.62

5.83

25.19

6.71

27.22

5.91

35.54

5.52

38.36

5.77

27.03

6.60

30.05

5.83

38.93

5.49

CRYOGENICS.

SEPTEMBER

1977

517